[Codeforces 897E. Willem, Chtholly and Seniorious]概率隨機+set

[Codeforces 897E. Willem, Chtholly and Seniorious]概率隨機+set

分類：math probability random

1. 題目鏈接

[Codeforces 897E. Willem, Chtholly and Seniorious]

2. 題意描述

有一個序列a[] ,長度爲n 。然後是m 次操作。
操作1：將al,al+1,…,ar 的數,全部加上x ;
操作2：將al,al+1,…,ar 的數,全部置位x ;
操作3：求al,al+1,…,ar 的數中的第x 小的數;
操作4：求(∑ri=laxi)%y ;
注意：序列a[] ，以及所有操作（即op,l,r,x,y ）都是等概率的隨機生成的。

數據範圍：
1 ≤ n, m ≤ 105, 0 ≤ seed < 109 + 7, 1 ≤ vmax ≤ 109

3. 解題思路

上面四種操作，出現的概率都是14 。
假如用線段來描述序列a[] 中值相同的子序列。當n=105 時，初始的時候，序列a[] 隨機分佈，所以線段很很多。但是經過多次操作2之後，會讓這個序列極速收斂，最終收斂到小於50 左右。然而概率論太差，我並不會證明合併到線段數目小於某一個值的次數的數學期望。
但是自己打印了一下，大概250 次合併就能讓線段數目小於200。
然後，知道了這種性質之後，就可以用set 來保存維護線段。剩下的操作就是暴力刪除一堆線段，暴力插入一些線段，暴力對線段進行查詢。
所以說，學好概率論太重要啦！

4. 實現代碼

#include <bits/stdc++.h>
using namespace std;

typedef long long ll;
typedef pair<int, int> pii;
typedef pair<ll, ll> pll;

const int inf = 0x3f3f3f3f;
const ll infl = 0x3f3f3f3f3f3f3f3fLL;
template<typename T> inline void umax(T &a, T b) { a = max(a, b); }
template<typename T> inline void umin(T &a, T b) { a = min(a, b); }
void debug() { cout << endl; }
template<typename T, typename ...R> void debug (T f, R ...r) { cout << "[" << f << "]"; debug (r...); }

const int MAXN = 100005;

ll n, m, seed, vmax, a[MAXN];

ll rnd() {
    ll ret = seed;
    seed = (seed * 7 + 13) % 1000000007;
    return ret;
}

struct Line {
    pii itv; ll val;
    Line() {}
    Line(pii itv, ll val) : itv(itv), val(val) {}
    bool operator < (const Line& line) const {
        return itv < line.itv;
    }
};

bool cmp_val(const Line& l1, const Line& l2)  {
    return l1.val < l2.val;
}

set<Line> st;
set<Line>::iterator it, it1, it2;
void oper1(int l, int r, ll x) {
    // [it1, it2)
    it1 = st.upper_bound(Line(pii(l, inf), 0ll)); --it1;
    it2 = st.upper_bound(Line(pii(r, inf), 0ll));
    vector<Line> buf;
    for (it = it1; it != it2; ++it) {
        if ((it->itv).first > r) break;
        buf.push_back(*it);
    }
    st.erase(it1, it2);
    if (buf[0].itv.first < l) {
        st.insert(Line(pii(buf[0].itv.first, l - 1), buf[0].val));
        buf[0].itv.first = l;
    }
    int sz = buf.size();
    if (buf[sz - 1].itv.second > r) {
        st.insert(Line(pii(r + 1, buf[sz - 1].itv.second), buf[sz - 1].val));
        buf[sz - 1].itv.second = r;
    }
    for (Line& line : buf) {
        line.val += x;
        st.insert(line);
    }
    // for(int i = l; i <= r; ++i) a[i] += x;
}

void oper2(int l, int r, ll x) {
    // [it1, it2)
    it1 = st.upper_bound(Line(pii(l, inf), 0ll)); --it1;
    it2 = st.upper_bound(Line(pii(r, inf), 0ll));
    vector<Line> buf;
    for (it = it1; it != it2; ++it) {
        if ((it->itv).first > r) break;
        buf.push_back(*it);
    }
    st.erase(it1, it2);
    if (buf[0].itv.first < l && buf[0].val != x) {
        st.insert(Line(pii(buf[0].itv.first, l - 1), buf[0].val));
        buf[0].itv.first = l;
    }
    int sz = buf.size();
    if (buf[sz - 1].itv.second > r && buf[sz - 1].val != x) {
        st.insert(Line(pii(r + 1, buf[sz - 1].itv.second), buf[sz - 1].val));
        buf[sz - 1].itv.second = r;
    }
    st.insert(Line(pii(buf[0].itv.first, buf[sz - 1].itv.second), x));
    // for(int i = l; i <= r; ++i) a[i] = x;
}

ll oper3(int l, int r, ll x) {
    // [it1, it2)
    it1 = st.upper_bound(Line(pii(l, inf), 0ll)); --it1;
    it2 = st.upper_bound(Line(pii(r, inf), 0ll));
    vector<Line> buf;
    for (it = it1; it != it2; ++it) {
        if ((it->itv).first > r) break;
        buf.push_back(*it);
    }
    if (buf[0].itv.first < l) {
        buf[0].itv.first = l;
    }
    int sz = buf.size();
    if (buf[sz - 1].itv.second > r) {
        buf[sz - 1].itv.second = r;
    }
    sort(buf.begin(), buf.end(), cmp_val);
    ll w = 0;
    for (Line& line : buf) {
        w += (line.itv.second - line.itv.first + 1);
        if (w >= x) return line.val;
    }
    return -1;
}

ll qpow(ll a, ll b, ll mod) {
    ll ret = 1;
    a = a % mod;    // notice here!!!
    while (b > 0) {
        if (b & 1) ret = ret * a % mod;
        a = a * a % mod;
        b >>= 1;
    }
    return ret;
}

ll oper4(int l, int r, ll x, ll y) {
    // [it1, it2)
    it1 = st.upper_bound(Line(pii(l, inf), 0ll)); --it1;
    it2 = st.upper_bound(Line(pii(r, inf), 0ll));
    vector<Line> buf;
    for (it = it1; it != it2; ++it) {
        if ((it->itv).first > r) break;
        buf.push_back(*it);
    }
    if (buf[0].itv.first < l) {
        buf[0].itv.first = l;
    }
    int sz = buf.size();
    if (buf[sz - 1].itv.second > r) {
        buf[sz - 1].itv.second = r;
    }

    ll ret = 0;
    for (Line& line : buf) {
        int w = (line.itv.second - line.itv.first + 1);
        ret += (ll)w * qpow(line.val, x, y) % y;
        ret %= y;
    }
    return ret;
}

int main() {
#ifdef ___LOCAL_WONZY___
    freopen("input.txt", "r", stdin);
    freopen("output.txt", "w+", stdout);
#endif // ___LOCAL_WONZY___
    cin >> n >> m >> seed >> vmax;
    for (int i = 1; i <= n; ++i) a[i] = (rnd() % vmax) + 1;
    st.clear();
    for (int i = 1; i <= n; ++i) {
        int j = i;
        while (j + 1 <= n && a[i] == a[j + 1]) ++j;
        st.insert(Line(pii(i, j), a[i]));
        i = j;
    }

    int op, l, r; ll x, y;
    int cnt = 0;
    for (int i = 1; i <= m; ++i) {
        op = rnd() % 4 + 1;
        l = rnd() % n + 1;
        r = rnd() % n + 1;
        if (l > r) swap(l, r);
        if (op == 3) x = (rnd() % (r - l + 1)) + 1;
        else x = (rnd() % vmax) + 1;
        if (op == 4) y = (rnd() % vmax) + 1;

        // debug(op, l, r, x, y);

        if (op == 1) oper1(l, r, x);
        else if (op == 2) oper2(l, r, x);
        else if (op == 3) {
            ll ret = oper3(l, r, x);
            cout << ret << endl;
        } else {
            ll ret = oper4(l, r, x, y);
            cout << ret << endl;
        }
    }
#ifdef ___LOCAL_WONZY___
    cout << "Time elapsed: " << 1.0 * clock() / CLOCKS_PER_SEC * 1000 << "ms." << endl;
#endif // ___LOCAL_WONZY___
    return 0;
}